Ribs and Fans Ribs and Fans of Bézier Curves and Surfaces Reporter: Dongmei Zhang 2007.11.21.
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Transcript of Ribs and Fans Ribs and Fans of Bézier Curves and Surfaces Reporter: Dongmei Zhang 2007.11.21.
Ribs and FansRibs and Fans of Bézier Curves and Surfaces
Reporter: Dongmei Zhang
2007.11.21
Papers (Joo-Haeng Lee and Hyungjun Park)
• Ribs and Fans of Bézier Curves and Surfaces Computer-Aided Design & Applications 2005
• Geometric Properties of Ribs and Fans of a Bézier Curve
CKJC 2006, Hangzhou
• A Note on Morphological Development and Transformation of Bézier Curves based on Ribs and Fans
SPM 2007, Beijing
Definition and Decomposition
Ribs and Fans
of Bézier Curves and Surfaces
Computer-Aided Design & Applications 2005
Ribs and Fans
curve surface
Ribs of a Bézier Curve
• A Bézier curve:
• Rib control points:
• Rib (a Bézier curve of degree k):
0
( ) ( )n
ni i
i
b t b B t
, 0,1,...ni ib r i n
1 11 , 0 , 1 1k k k
i i i
k i ir r r i k k n
k k
0
( ) ( ), 1 1k
k k ki i
i
r t r B t k n
1k
ir
11kir
12kir
11kir
Examples (cubic Bézier curve)
1 11 , 0
k k ki i i
k i ir r r i k
k k
A cubic Bézier curve
Control points of ribs Rib curves
2 3 2 3 3 2 30 0 1 1 2 2 3
1, ( ),
2r r r r r r r
Especial property
0 0 , , 1k n k nk nr r r r k n
Fans of a Bézier Curve
• A Bézier curve:
• Fan control vectors:
• Fan (“a Bézier curve” of degree k):
0
( ) ( )n
ni i
i
b t b B t
, 0,1,...ni ib r i n
2 2 21 2
1( ), 0 , 0 22
k k k ki i i if r r r i k k n
0
( ) ( ), 0 2k
k k ki i
i
f t f B t k n
1k
ir
11kir
12kir
11kir
Examples (cubic Bézier curve)
Control vectors Fans
2 2 21 2
1( ), 02
k k k ki i i if r r r i k
1 3 3 30 1 0 2
1( )2
f r r r
1 3 3 31 2 1 3
1( )2
f r r r
0 2 2 20 1 0 2
1( )2
f r r r
Decomposition
• Theorem: A Bézier curve of degree n can be decomposed into a rib of degree n-1 and a fan of degree n-2.
1 2( ) ( ) ( ) 2 (1 ) ( )n n nb t r t r t t t f t
Proof (mathematical induction)
• Base step: n=21 2 1 20 0 1 2,r r r r
11 1 1 2 2
0 20
( ) ( ) (1 )i ii
r t r B t t r tr
0 2 2 20 1 0 2
1( )2
f r r r 0 0 2 2 20 1 0 2
1( ) ( )
2f t f r r r
1 0
2 2 2 2 20 2 1 0 2
2 2 2 2 20 1 2
2
( ) 2 (1 ) ( )
1(1 ) 2 (1 )( ( ))
2
(1 ) 2 (1 )
( )
r t t t f t
t r tr t t r r r
t r t t r t r
r t
Proof
• Induction hypothesis (n=k):
• n=k+1 :
1 2( ) ( ) 2 (1 ) ( )k k kr t r t t t f t
1 1( ) ( ) 2 (1 ) ( )k k kr t r t t t f t
11 1 1 11 2
0
1( ( )) ( )
2
kk k k ki i i i
i
r r r B t
1
1 1 1
0
( ) ( )k
k k ki i
i
f t f B t
Proof
0
( ) ( )k
k k ki i
i
r t r B t
1 11
0
1(( ) ) ( )
kk k ki i i
i
k i r ir B tk
1 11
0 0
1 1( ) ( ) ( )
k kk k k ki i i i
i i
k i r B t ir B tk k
1 1
1( ) (1 ) ( ), ( ) ( ), 0n n n ni i i i
n nB t t B t B t tB t i n
n i i
11 1 1 1
1 10 1
1 1( ) (1 ) ( ) ( )
k kk k k ki i i i
i i
kk i r t B t ir tB t
k k i k
1 11 1 1 1
20 0
(1 ) ( ) ( )k k
k k k ki i i i
i i
r t B t ir tB t
1
1 1 12
0
((1 ) ) ( )k
k k ki i i
i
t r tr B t
1( ) 2 (1 ) ( )k kr t t t f t 1( )kr t
Proof1( ) 2 (1 ) ( )k kr t t t f t
1 11 1 1 1 1 1 1
2 1 20 0
1((1 ) ) ( ) 2 (1 ) ( ( )) ( )
2
k kk k k k k k ki i i i i i i
i i
t r tr B t t t r r r B t
1
2 1 1 2 1 11 2
0
((1 ) 2 (1 ) ) ( )k
k k k ki i i i
i
t r t t r t r B t
1 2
1 2 1
0 0
( ( )) ( )k
k ki j j i
i j
r B t B t
2 2 21 2 1 1 2 1 1 2 1
0 0 1 10 0 0
( ( ) ( )) ... ( ( ) ( )) ... ( ( ) ( ))k k k k k kj j i j i j k j k j
j j j
r B t B t r B t B t r B t B t
1 2
1 2 1
0 0
( ( ) ( ))k
k ki j i j
i j
r B t B t
1
1( ) (1 ) ( ) ( )n n ni i iB t t B t tB t
2
1 2 1 1 2 1 2 11 2
0
( ) (1 ) ( ) 2 (1 ) ( ) ( ) ( ) ( )n n n n ni i i i j i j
j
B t t B t t t B t t B t B t B t
1( )kr t
Decomposition
• Theorem: A Bézier curve of degree n can be decomposed into a rib of degree n-1 and a fan of degree n-2.
1 2( ) ( ) ( ) 2 (1 ) ( )n n nb t r t r t t t f t
Decomposition
• Corollary: A Bézier curve of degree n can be decomposed into a single rib of degree l and a sequence of n-l fans of degrees from n-2 to l-1.
2
1
( ) ( ) ( ) 2 (1 ) ( ),
1 1
nn l k
k l
b t r t r t t t f t
l n
Surface case, ,
,0 0
( , ) ( ) ( )m n
m n m n m ni j i j
i j
r u v r B u B v
, , , ,
0, , *, ,0
{ ,..., } ( ) ( )m
m n m n m n m n mj m j j i j i
i
r r r u r B u
i 1i
2i 1i
1j
j
1j
2j
, , , ,,0 , ,* ,
0
{ ,..., } ( ) ( )n
m n m n m n m n ni i n i i j j
j
r r r v r B v
, 1, 2,*, *, *,( , ) ( , ) 2 (1 ) ( , )m n m n m nj j jr u v r u v u u p u v
1, , ,, , 1,
1(( 1 ) ), 0 11
m n m n m ni j i j i jr m i r ir i m
m
2, , , ,, 1, , 2,
1( ), 0 22
m n m n m n m ni j i j i j i jp r r r i m
, , 1 , 2,* ,* ,*( ) ( ) 2 (1 ) ( )m n m n m ni i ir v r v v v q v
, 1 , ,, , , 1
1(( 1 ) ), 0 11
m n m n m ni j i j i jr n j r jr j n
n
, 2 , , ,, , 1 , , 2
1( ), 0 22
m n m n m n m ni j i j i j i jq r r r j n
Composite transformation
Rib and its control points, ,
,0 0
( , ) ( ) ( )m n
m n m n m ni j i j
i j
r u v r B u B v
i 1i
2i 1i
1j
j
1j
2j
, 1, 1 1, 1, , 1,
1, 1 1, 1, 1 1, 1
1(( )(( ) )
(( ) ))
1 ,1
k l k l k li j i j i j
k l k li j i j
r l j k i r irkl
j k i r ir
k n l m
,
, , , ,0,0 0,0 ,0 ,0
, , , ,0, 0, , ,
,
,
k l m n k l m nk m
k l m n k l m nl n k l m n
r r r r
r r r r
Fan and its control vectors, ,
,0 0
( , ) ( ) ( )m n
m n m n m ni j i j
i j
r u v r B u B v
, 1, 2 1, 2 2, 1, , 1 , , 2
1, 2 1, 2 1, 21, 1 1, 1, 2
1 1(( )( ( ))
21
( ( ))),2
1 1,0 2
k l k l k l k li j i j i j i j
k l k l k li j i j i j
q k i r r rk
i r r r
k m l n
i 1i
2i 1i
1j
j
1j
2j , 2, 2 2, 2 2, 2, 1, 1 , 1 2, 1
2, 2 2, 2 2, 21, , 2,
2, 2 2, 2 2, 21, 2 , 2 2, 2
1( ( ))
21 1(( ( ))2 2
1( ( ))),
20 2,0 2
k l k l k l k li j i j i j i j
k l k l k li j i j i j
k l k l k li j i j i j
f r r r
r r r
r r r
k m l n
Decomposition
• Theorem: A Bézier surface of degree (m,n) can be decomposed into a rib of degree (m-1,n-1) and three fans.
, 1, 1 2, 1 1, 2
2, 2
1, 1 2, 2
( , ) ( , ) 2 (1 ) ( , ) 2 (1 )
4 (1 )(1 ) ( , )
( , ) ( , )
m n m n m n m n
m n
m n m n
r u v r u v u u p u v v v q
uv u v f u v
r u v u v
Proof
, ,,
0 0
( , ) ( ) ( )m n
m n m n m ni j i j
i j
r u v r B u B v
固定 v
在 u方向上
固定 u
在 v方向上, 1, 1 2, 2( , ) ( , ) ( , )m n m n m nr u v r u v u v
Decomposition
• Corollary: A Bézier surface of degree (m,n) can be decomposed into a single rib of degree (m-k,n-k) and a sequence of k composite fans.
, , 1 , 1
1
( , ) ( , ) ( , ),
1 ( , ) 1
km n m k n k m i n i
i
r u v r u v u v
k MIN m n
Example(bi-cubic Bézier surface)
Example(bi-cubic Bézier surface)
Examples (Bézier curve of degree 9)
• d
Examples (Bézier curve of degree 10)
Geometric Properties
Geometric Properties
of Ribs and Fans of a Bézier Curve
CKJC 2006, Hangzhou
• Composite fans
• Rib-invariant deformation
Composite fans
• Property 1: A Bézier curve of degree n can be composed into a straight line segment and a composite fan of degree n-2.
22 2 21 2
0
( ) ( ) 2 (1 ) ( ), ( ) ( )nn n nn n
iii
r t r t t t f t f t f B t
21
0
( ) ( ) 2 (1 ) ( )n
n r
r
r t r t t t f t
0
( ) ( )r
r r ri i
i
f t f B t
degree elevation
2 21 2
0
( ) ( ) 2 (1 ) ( )n nn n
iii
r t r t t t f B t
Composite fans
• Property 2: A straight line segment and a composite fan of degree n-2 can build a unique Bézier curve of degree n.
21
0
21 10 1 1
( ) 2 (1 ) ( ) ( ),
1 1 2 ( )( )
1 ( 1)
nn n ni i
i
nni i
r t t t f t r B t
n n n i n ir r r f
i i i n n
Proof
,1( )
nr t1( )r t
21
( 1)( 1)(1 ) ( ) ( )
( 1)( 2)n ni i
i n it t B t B t
n n
2 2 2
0
2 (1 ) ( )n n n
iii
t t f B t
0
* ( )n
ni
i
B t
21
0
21 10 1 1
( ) 2 (1 ) ( ) ( ),
1 1 2 ( )( )
1 ( 1)
nn n ni i
i
nni i
r t t t f t r B t
n n n i n ir r r f
i i i n n
degree elevation
Rib-invariant Deformation
• Property 3:
For a given Bézier curve of degree n, we can modify up to n-d control points while preserving a rib of degree d.
Moreover, if we specify n-d control points explicitly, we can determine the unknown d-1 control points uniquely.
Proof
• Initial Bézier curve:
• Rib of degree d:
• New Bézier curve:
0 0
, 0,1...,n n
j n ji i i i
i i
b l r l j d
0
( ) ( ), 0,1,...n
n n ni i
i
r t r B t i n
0
, 0,1...,n
d n jj i i
i
r r l j d
0
( ) ( )n
ni i
i
b t b B t
Example (quartic Bézier curve)
Example (curve of degree 9)
Applications
A Note on Morphological Development and Transformation of Bézier
Curves
based on Ribs and Fans
SPM 2007, Beijing
Morphological development
• To find a sequence of shapes that believed to represent a pattern of growth.
Morphological transformation
• To find a sequence curves that represents the pattern from one curve to another.
Morphological development
• Current shape:• Initial shape (simple, minimum features):• Developmental pattern:
( ) ( )nb s r s1( )r s
1( , ) ( ) ( , )d s t r s p s t
1( ,0) ( ), ( ,1) ( )nd s r s d s r s
DCF (development by composite fan)
• Linear development:1
,
21
1
( , ) (1 ) ( ) ( )
( ) (2 (1 ) ( ))
( ) ( , )
nc f
n
cf
d s t t r s tr s
r s t s s f s
r s p s t
, ( , ; , )
(1 ) ( ) ( )
( ) ( , )
n qc f
q n
qcf
d s t r r
t r s tr s
r s p s t
DCF(development by composite fan)
DFL(development by fan lines)
• Utilize each rib:1( , ; , ) ( ) ( )
nn q i
fl i qi q
d s t r r r s N t
DFL(development by fan lines)
DFL(development by fan lines)
DSC(development by spline curves)
• path: a smooth curve. ( , ; , ) ( ) ( )n
n q i n qsc i q
i q
d s t r r r s B t
DSC (development by spline curves)
Comparision
Morphological transformation
• Three methods (TLI,TCE,TDE).
• TLI (Transformation by linear interpolation). correspondence: index of control points.
TCE(by cubic blending and extrapolation)
• Two Bézier curves:• Lower ribs:
3 2 20 0 1( , ) (1 ) ( , ) ( , )
( , )c
t s t t t s t tt s t
t s t
( ), ( )m na br s r s
( ), ( )p qa br s r s
3 3 3 30 1 2 3( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )m p q n
c a a b bt s t B t r s B t r s B t r s B t r s
10
11
12
( , ) (1 ) ( ) ( )
( , ) (1 ) ( ) ( )
( , ) (1 ) ( ) ( )
m pa a
p qa b
q nb b
t s t t r s tr s
t s t t r s tr s
t s t t r s tr s
2 1 10 0 1
2 1 11 1 2
( , ) (1 ) ( , ) ( , )
( , ) (1 ) ( , ) ( , )
t s t t t s t tt s t
t s t t t s t tt s t
1 31 0( , ) (1 ( )) ( , ) ( ) ( , )
( ) 1 (1 3 (1 ))cet s t e t t s t e t t s t
e t t t
TDE(by development and extrapolation)
10
11
12
( , ) (1 ) ( ) ( )
( , ) (1 ) ( ) ( )
( , ) (1 ) ( ) ( )
m pa a
p qa b
q nb b
t s t t r s tr s
t s t t r s tr s
t s t t r s tr s
2 1 2 2, 0 1 1 2( , ) ( , ; , ) ( ) ( , ) ( ) ( , ; , ) ( )m nd sc sc a sc bt s t d s t r p B t t s t B t d s t r q B t
3 1 2 1 2 1 20 0 0 1 1 2 2( , ) ( , ) ( ) ( , ) ( ) ( , ) ( )t s t t s t B t t s t B t t s t B t
2 1 2 2, 0 1 1 2( , ) ( , ; , ) ( ) ( , ) ( ) ( , ; , ) ( )m nd fl fl a fl bt s t d s t r p B t t s t B t d s t r q B t
1,* 1 ,*( , ) (1 ( )) ( , ) ( ) ( , )de dt s t e t t s t e t t s t
( ) 1 (1 (1 ))e t t t
( ) 1 (1 3 (1 ))e t t t
Comparision
Summary
• Definition: Ribs and Fans
• Decomposition. Bézier curve = rib + scaled fans.
• Property. Rib + composite fan= Bézier curve=rib.
Rib-invariant deformation.
• Morphological applications Development and transformation.
Thanks